We consider the prediction model to estimate \(Y\)  as \(\hat{Y} = \alpha _1 X_1 + \alpha _2 X_2 + \alpha _3 X_3\) . Then. it follows that if \(\alpha _3 > 0\) , the regression coefficients will explicitly depend on \(\sigma (e)\)  and the OOD risk of the model can grow without bound, as noted in [1]}. Within the IRM framework, the only feasible representation \(\Phi \)  that yields invariant predictors across \((E, e)\)  is \(\Phi (X) = X_1\)  and the corresponding regression coefficients are \((1, 0, 0)\)  respectively. While this solution works and generalizes for OOD tasks w.r.t. \(e\) , we lose in terms of performance by discarding \(X_2\) . In addition, deviating from the IRM solution via IRMv1 (as in the previous section) is not feasible since we require invariance w.r.t \(e\)  for a finite risk in worst-case environments.
